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Abstract

The coupling of mechanical and optical degrees of freedom via radiation pressure has been a subject of early research in the context of gravitational wave detection. Recent experimental advances have allowed studying for the first time the modifications of mechanical dynamics provided by radiation pressure. This paper reviews the consequences of back-action of light confined in whispering-gallery dielectric micro-cavities, and presents a unified treatment of its two manifestations: notably the parametric instability (mechanical amplification and oscillation) and radiation pressure back-action cooling. Parametric instability offers a novel “photonic clock” which is driven purely by the pressure of light. In contrast, radiation pressure cooling can surpass existing cryogenic technologies and offers cooling to phonon occupancies below unity and provides a route towards cavity Quantum Optomechanics

The two manifestations of dynamic back-action: blue-detuned and red-detuned pump wave (green) with respect to optical mode line-shape (blue) provide mechanical amplification and cooling, respectively. Also shown in the lower panels are motional sidebands (Stokes and anti-Stokes fields) generated by mirror vibration and subsequent Doppler-shifts of the circulating pump field. The amplitudes of these motional sidebands are asymmetric owing to cavity enhancement of the Doppler scattering process.

Work done during one cycle of mechanical oscillation can be understood using a PV diagram for the radiation pressure applied to a piston-mirror versus the mode volume displaced during the cycle. In this diagram the cycle follows a contour that circumscribes an area in PV space and hence work is performed during the cycle. The sense in which the contour is traversed (clockwise or counterclockwise) depends upon whether the pump is blue or red detuned with respect to the optical mode. Positive work (amplification) or negative work (cooling) are performed by the photon gas on the piston mirror in the corresponding cases.

Dynamics in the weak retardation regime. Experimental displacement spectral density functions for a mechanical mode with eigenfrequency 40.6 MHz measured using three, distinct pump powers for both blue and red pump detuning. The mode is thermally excited (green data) and its linewidth can be seen to narrow under blue pump detuning (red data) on account of the presence of mechanical gain (not sufficient in the present measurement to excite full, regenerative oscillations); and to broaden under red pump detuning on account of radiation pressure damping (blue data).

The mechanical amplification and cooling rate as a function of detuning and normalized mechanical frequency. Also shown is the optimum amplification and cooling rate for fixed frequency (dotted lines). In the simulation, pump power and cavity dimension are fixed parameters.

Dynamics in the regime where Ωm>κ as reported in reference [28]. Upper panel shows the induced damping/amplification rate (δm=Γeff/2π) as a function of normalized detuning of the laser at constant power. The points represent actual experiments on toroidal microcavities, and the solid line denotes a fit using the sideband theoretical model (Equations 16 and 20). Lower panel shows the mechanical frequency shift as a function of normalized detuning. Arrow denotes the point where the radiation pressure force is entirely viscous causing negligible in phase, but a maximum quadrature component. The region between the dotted lines denotes the onset of the parametric instability (as discussed in section 4). Graph stems from reference [28].

Upper panel: SEM images and mechanical modes of several types of whispering gallery mode microcavities: toroid microcavities [60] microdisks [59] and microspheres [58]. Also shown are the stress and strain field in cross section of the fundamental radial breathing modes, which include radial dilatation of the cavity boundary. Lower panel: the dispersion diagram for the lowest lying, rotationally symmetric mechanical modes for a toroid (as a function of its undercut) and for a microsphere (as a function of radius).

Scanning probe microscopy of the two lowest lying micro-mechanical resonances of a toroid microcavity. Lower graph: The normalized mechanical frequency shift for the first mode as a function of position. Upper graph: The normalized frequency shift for the second mechanical mode as a function of scanned distance across the toroid. Superimposed is the scaled amplitude (solid line) and the amplitude squared (dotted line) of the mechanical oscillator modes obtained by finite element simulation of the exact geometry parameters (as inferred by SEM).

Calibrated [47] displacement spectral density as measured by the setup shown in Figure 10. The peaks denote different mechanical eigenmodes of the toroidal microcavity. The probe power is sufficiently weak such that the mechanical modes amplitude is dominated by Brownian motion at room temperature and backaction effects are negligible. Cross-sectional representations of the n=1,2,3 modes and their corresponding spectral peaks are also given as inferred by finite element simulations.

Back-action tuning for mode selection with a fixed laser detuning corresponding to Δ=κ/2. The target mode that receives maximum gain (or optimal cooling for Δ=-κ/2) can be controlled by setting the cavity linewidth to produce maximum sideband asymmetry for that particular mechanical mode. In this schematic, three mechanical modes (having frequencies Ωm,i, i=1,2,3) interact with an optical pump, however, in the present scenario only the intermediate mode experiences maximum gain (or cooling) since its sideband asymmetry is maximal (since Ωm,2/κ=0.5). It is important to note however, that if the laser detuning is allowed to vary as well, the highest frequency mode would experience the the largest gain if Δ=Ωm,2 was chosen.

Main figure: The observed threshold for the parametric oscillation (of an n=1 mode) as a function of inverse mechanical quality factor. In the experiment, variation of Q factor was achieved by placing a fiber tip in mechanical contact with the silica membrane, which thereby allowed reduction of the mechanical Q (cf. inset). The mechanical mode was a 6 MHz flexural mode.

Main panel shows the measured mechanical oscillation threshold (in micro-Watts) from Ref. [22] plotted versus the optical Q factor for the fundamental flexural mode (n=1, Ωm/2π=4.4MHz, meff≈3.3×10-8 kg, Qm≈3500). The solid line is a one-parameter theoretical fit obtained from the minimum threshold equation by first performing a minimization with respect to coupling (C) and pump wavelength detuning (D), and then fitting by adjustment of the effective mass. Inset: The measured threshold for the 3rd order mode (n=3, Ωm/2π=49MHz, meff≈5×10-11 kg, Qm≈2800) plotted versus optical Q. The solid line gives again the theoretical prediction. The n=1 data from the main panel is superimposed for comparison. Figure stems from reference [22].

Line-width measurements from Ref. [67] of the opto-mechanical oscillator for different amplitudes of oscillation plotted in picometers. The measurement is done at room temperature (dots) and at temperature 90 °C above room temperature (stars). The solid lines and the corresponding equations are the best fits to the log-log data. Solid line denotes theoretically expected behavior.

Main figure shows the normalized, measured noise spectra around the mechanical breathing mode frequency for Δ·τ≈-0.5 and varying power (0.25,0.75,1.25, and 1.75 mW). The effective temperatures were inferred using mechanical damping, with the lowest attained temperature being 11K. (b) Inset shows increase in the linewidth (effective damping δm=Γeff/2π) of the 57.8-MHz mode as a function of launched power, exhibiting the expected linear behavior as theoretically predicted. From reference [28].

The frequency response from 0–200MHz of a toroidal opto-mechanical system, adopted from Reference [28]. The plateau occurring above 1MHz is ascribed to the (instantaneous) Kerr nonlinearity of silica (dotted line). The high-frequency cutoff is due to both detector and cavity bandwidth. The response poles at low frequency are thermal in nature. Inset: Data in the vicinity of mechanical oscillator response shows the interference of the Kerr nonlinearity and the radiation pressure-driven micromechanical resonator (which, on resonance, is π/2 out-of phase with the modulating pump and the instantaneous Kerr nonlinearity). From the fits (solid lines) it can be inferred that the radiation pressure response is a factor of 260 larger than the Kerr response and a factor of ×100 larger than the thermo-mechanical contribution.